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a89d282 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 | import OAI.Probability.InvariantIsing.Core.KernelRelativeEntropy
import OAI.Probability.InvariantIsing.Core.TiltedRelativeEntropy
namespace OAI
/-! Nonnegative entropy budgets for successive finite sampling steps. -/
noncomputable section
open MeasureTheory ProbabilityTheory InformationTheory IsingPerceptron
open scoped ENNReal
namespace InvariantIsing
lemma entropy_budget_of_real {X : Type*} [MeasurableSpace X]
(μ ν : Measure X) (D : X → ℝ) (hD : Integrable D μ) (hD0 : ∀ x, 0 ≤ D x)
{c : ℝ} (hK : klDiv μ ν ≠ ⊤) (h : (klDiv μ ν).toReal + (∫ x, D x ∂μ) ≤ c) :
klDiv μ ν + (∫⁻ x, ENNReal.ofReal (D x) ∂μ) ≤ ENNReal.ofReal c := by
rw [← ofReal_integral_eq_lintegral_ofReal hD (Filter.Eventually.of_forall hD0),
← ENNReal.ofReal_toReal hK, ← ENNReal.ofReal_add ENNReal.toReal_nonneg
(integral_nonneg hD0)]
exact ENNReal.ofReal_le_ofReal h
lemma entropy_budget_drop {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y]
[MeasurableSpace.CountableOrCountablyGenerated X Y]
(μ ν : Measure X) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν]
(κ η : Kernel X Y) [IsMarkovKernel κ] [IsMarkovKernel η]
(hac : ∀ x, κ x ≪ η x) (D : X → ℝ≥0∞)
(h : ∀ x, kernelRelativeEntropy κ η x ≤ D x) :
klDiv (κ ∘ₘ μ) (η ∘ₘ ν) ≤ klDiv μ ν + ∫⁻ x, D x ∂μ :=
(klDiv_comp_le_integral μ ν κ η hac).trans (add_le_add_right (lintegral_mono h) _)
lemma entropy_budget_map {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y]
(μ ν : Measure X) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν]
(f : X → Y) (hf : Measurable f) (D : Y → ℝ≥0∞) (hD : Measurable D) :
klDiv (μ.map f) (ν.map f) + (∫⁻ y, D y ∂μ.map f) ≤
klDiv μ ν + ∫⁻ x, D (f x) ∂μ := by
rw [lintegral_map hD hf]
exact add_le_add_left (klDiv_map_le μ ν hf) _
lemma tilted_pair_entropy_budget_ennreal {X : Type*} [MeasurableSpace X]
(μ : Measure X) [IsProbabilityMeasure μ] (F G : X → ℝ)
{b : ℝ} (hb : 0 < b) (hb1 : b ≤ 1)
(hF : Integrable (fun x => Real.exp (b * F x)) μ)
(hG : Integrable (fun x => Real.exp (b * G x)) μ)
(hFG : Integrable (fun x => F x - G x) (μ.tilted (fun x => b * G x)))
(hGF : ∀ x, G x ≤ F x) :
klDiv (μ.tilted (fun x => b * G x)) (μ.tilted (fun x => b * F x)) +
(∫⁻ x, ENNReal.ofReal (F x - G x) ∂μ.tilted (fun x => b * G x)) ≤
ENNReal.ofReal (logMean b μ F - logMean b μ G) :=
entropy_budget_of_real _ _ _ hFG (fun x => sub_nonneg.mpr (hGF x))
(tilted_pair_entropy_identity μ F G hb hF hG hFG).1
(tilted_pair_entropy_budget μ F G hb hb1 hF hG hFG)
end InvariantIsing
end
end OAI
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